ON PLANE PARTITIONS and N–COLOR PARTITIONS Thesis

ON PLANE PARTITIONS AND n–COLOR PARTITIONS

Thesis submitted in partial fulﬁllment of the requirement for the award of the degree of

Master of Science

Mathematics and Computing

Submitted by

Amrinder Kaur Roll no. 301503002

Under the guidance of

Dr. Meenakshi Rana

July 2017 School Of Mathematics Thapar University, Patiala INDIA

Acknowledgements

First of all, I would like to express my gratitude to Dr. Meenakshi Rana, Associate Professor, SOM, Thapar University, Patiala for their patient guidance and support throughout the work. I am truly very fortunate to work under her as a student. I would like to thank Dr. A.K. Lal, Associate Professor and Head, SOM, Thapar University, Patiala for providing help and all the necessary facilities in the department and directly or indirectly encouraging me to work harder during the whole work. I thank my parents for their lovely support. I admire my parent’s determi- nation and sacriﬁce to get the best for me.

July, 2017 Amrinder Kaur

2 Abstract

The ﬁrst chapter is devoted to preliminaries and provides introduction to plane partitions and n–color partitions. The second chapter elaborates the bijection between plane partitions and n– color partitions through Bender and Knuth matrices using which some basic results are proved. The third chapter includes some theorems which give a nice interaction between the plane partition and n–color partition. A relation between the rows of the plane partition and the subscripts of the n–color partition of any number ν is established in this chapter. Our results give a simpler way for ﬁnding the corresponding plane partition for a given n–color partition and vice–versa.

3 Contents

1 Introduction 5 1.1 Partition ...... 5 1.2 Plane Partitions ...... 6 1.3 n–Color Partitions ...... 8 1.4 BKν–matrices ...... 8 1.5 Ei,j matrices ...... 9 1.6 Rogers–Ramanujan Identities ...... 9 1.7 Remark ...... 10

2 Bijection Between Plane Partition And n–Color Partition 11 2.1 Introduction ...... 11 2.2 Bijection ...... 11 2.3 Basic series and its Combinatorics ...... 14 2.4 Conclusion ...... 19

3 On Extensions Of Relations Between Plane Partitions and n–Color Partitions 20 3.1 Introduction ...... 20 3.2 Main Results ...... 20 3.3 Applications of Plane partitions ...... 25 3.3.1 Diamond Partitions ...... 25 3.3.2 Solid Partitions ...... 27

4 Chapter 1

Introduction

1.1 Partition

Deﬁnition 1.1.1 [8] A Partition of a positive integer n is a ﬁnite non increasing sequence of positive integers

a1 ≥ a2 ≥ ...... ≥ ar such that r X ai = n. i=1

0 The ais are called the parts or summands of the partition. We denote by p(n) the number of partitions of n. p(0) = 1 as zero has one partition which is the empty partition(It has no parts). The Partitions of n = 4 are

4, 3+1, 2+2, 2+1+1, 1+1+1+1.

Hence p(4) = 5. In the deﬁnition of partitions, the order does not matter. So 4+3 and 3+4 are the same partitions of 7. Thus a partition is an unordered collection of parts. An ordered collection is called a Composition. Thus 4+3 and 3+4 are two diﬀerent compositions of 7. The generating function of p(n) is given by

∞ X 1 p(n)qn = , (q; q) n=0 ∞

5 where |q| < 1 and (q; q)n is a rising q–factorial deﬁned by ∞ Y (1 − aqi) (a; q) = n (1 − aqn+i) i=0 for any constant a. If n is a positive integer, then

n−1 (a; q)n = (1 − a)(1 − aq) ··· (1 − aq ) 2 and (a; q)∞ = (1 − a)(1 − aq)(1 − aq ) ··· An improved deﬁnition of partition regards the parts of the partition as being placed at points of a line •≥•≥•≥•≥•≥• and the symbol ≥ as regulating the magnitude of the parts at any two ad- jacent points. It is important to realise that the partitions may be regarded as partitions on a line or in one dimension of space.

1.2 Plane Partitions

Deﬁnition 1.2.1 [17] A Plane Partition of n is an array

n1,1 n1,2 n1,3 ··· n2,1 n2,2 n2,3 ··· ...... of non negative integers for which X nij = n, i,j where the rows and columns are in non increasing order:

nij ≥ n(i+1)j , nij ≥ ni(j+1) ∀ i, j ≥ 1.

If nij = 0 for all i > r, it is an r–rowed partition and if nij = 0 for all j > c, it is a c–columned partition. If nij ≤ m ∀ i, j ≥ 1, we say the parts do not exceed m. The generating function for plane partitions is

∞ ∞ X Y 1 P l(n)qn = , (1 − qn)n n=0 n=1

6 where P l(n) denotes the number of plane partitions of n. The six plane partitions of n = 3 are

3, 2 1, 2, 1 1 1, 1 1, 1. 1 1 1 1 Plane partitions are the partitions in two dimension of space. It is interesting to observe that the Ferrers–Sylvester graph of a partition of a unipartite number is in reality a partition in two dimensions. Such a graph of the partition 4 3 1 1 is •••• ••• • • For many purposes it is advantageous to replace the nodes by units, leading to the unit graph 1 1 1 1 1 1 1 1 1 This is a two–dimensional partition of the number 9 in agreement with the deﬁnition which is now given. Consider the points of a two–dimensional lattice

O ••••• x ••••• ••••• ••••• y and let the first row and first column be axes of x and y respectively. Suppose the parts of the partition to be placed at these points so that a descending order of magnitude is in evidence in each row in direction of the x–axis and also in each column in direction of the y–axis. The arrangement of numbers thus reached is defined to be a two–dimensional partition of the number par- titioned. Clearly the unit graph of a partition is a two–dimensional partition in which the part magnitude is limited not to exceed unity. In the case of a partition in two dimensions we are concerned with three limiting numbers, for we may limit

1. the number of rows,

7 2. the number of columns,

3. the part magnitude.

1.3 n–Color Partitions

Deﬁnition 1.3.1 [2] An n–color partition (also called a partition with “n copies of n”) of a positive integer ν is a partition in which a part of size n can come in n diﬀerent colors denoted by subscripts n1, n2, ··· , nn and the parts satisfy the order :

11 < 21 < 22 < 31 < 32 < 33 < 41 < 42 < 43 < 44 < ···

If P (ν) denotes the number of n–color partitions of ν, then the generating function is ∞ ∞ X Y 1 P (ν)qν = . (1 − qn)n ν=0 n=1 The n–color partitions of ν = 3 are

31, 32, 33, 2111, 2211, 111111

1.4 BKν–matrices

Definition 1.4.1 [6] For every non–negative integer ν, BKν–matrices(BK for Bender and Knuth) are defined as infinite matrices [ai,j] (i,j ≥ 1) of non–negative integer entries which satisfy X X r ( ai,j ) = ν r≥1 i+j=r+1

These are inﬁnite matrices but will be represented in the sequel by the largest possible square matrices whose last row(column) is non–zero.

Thus, for example, the six relevant BK3–matrices are given by

0 0 1 0 0 0 1 0 1 1 0 0 3 , , , , 0 0 0 , 0 0 0 . 1 0 0 0 1 1     0 0 0 1 0 0

8 1.5 Ei,j matrices

Definition 1.5.1 [1] We define a matrix Ei,j as an infinite matrix whose th (i, j) entry is 1 and all the other entries are zeroes. We call Ei,j distinct units of BKν–matrix.

For example, 0 0 0 0 1 E = 0 0 1 and E = 23   12 0 0 0 0 0

1.6 Rogers–Ramanujan Identities

A series involving factors like rising q–factorial (a; q)n deﬁned by

∞ Y (1 − aqi) (a; q) = n (1 − aqn+i) i=0 is called a basic series (or q-series, or Eulerian series). The following two “sum–product” basic series identities are known as the Rogers–Ramanujan identities

∞ 2 ∞ X qn Y 1 = , (1.1) (q; q) (1 − q5n−1)(1 − q5n−4) n=0 n n=1 ∞ 2 ∞ X qn +n Y 1 = . (1.2) (q; q) (1 − q5n−2)(1 − q5n−3) n=0 n n=1 They were ﬁrst discovered by Rogers [12] and rediscovered by Ramanujan in 1913. MacMahon [17] gave the following partition theoretic interpretations of (1.1) and (1.2), respectively: Theorem 1.6.1 The number of partitions of n into parts with the minimal diﬀerence 2 equals the number of partitions of n into parts ≡ ±1 (mod 5).

Theorem 1.6.2 The number of partitions of n with minimal part 2 and minimal diﬀerence 2 equals the number of partitions of n into parts ≡ ±2 (mod 5).

Partition theoretic interpretations of many more q–series identities like (1.1) and (1.2) have been given by several mathematicians. See, for instance, G¨ollnitz[10, 11], Gordon [5], Connor [19], Hirschhorn [13], Subbarao [15], Subbarao and Agarwal [16].

9 1.7 Remark

In the next chapter, we discuss the bijection between plane partitions and color partitions through a pair (S,T) of plane partitions. Each pair corresponds to a unique color partition and plane partition using BKν and Ei,j matrices discussed in this chapter. Then we study some Rogers–Ramanujan type identities using BKν matrices.

10 Chapter 2

Bijection Between Plane Partition And n–Color Partition

2.1 Introduction

In [6], a one to one correspondence between plane partitions of ν and BKν– matrices is established(also elaborated in [4]). And there exists a bijection between BKν–matrices and n–color partitions of ν which is given in [1]. The results given in Chapter 3 use these bijections. Hence it is imperative to illustrate this bijection, given in section 2.2.

2.2 Bijection

Consider an n–color partition of ν

m1n1 + m2n2 + m3n3 + ··· + mtnt ,

where ms is the part and ns is the subscript. Each single part msns of an n–color partition of ν can be mapped to a single part Ep,q as

ψ : mi → Ei,m−i+1 (2.1) and the inverse is possible as

−1 ψ : Ep,q → (p + q − 1)p

11 0 Using these Ep,q s, we construct a matrix A as given below

A = a1,1E1,1 +a1,2E1,2 +···+a2,1E2,1 +a2,2E2,2 +···+a3,1E3,1 +a3,2E3,2 +···

0 where ap,q s are non–negative integers which denote the multiplicities of 0 Ep,q s. From this matrix A, we ﬁrst construct a two line array σ1(A) as given below: Suppose ap,q = k > 0. Then enter k copies of p in the ﬁrst row of σ1(A) and k copies of q in the second row. For example, if we start with

1 0 2 A = 0 2 0 1 0 0

1 1 1 2 2 3 σ (A) = 1 1 3 3 2 2 1

Second, we permute the columns of σ1(A) so that (a) the elements of the ﬁrst row are in non–increasing order. (b) within a block of constancy of the ﬁrst row, the corresponding elements of the second row are in non–increasing order. This yields σ2(A). In the previous example,we have

3 2 2 1 1 1 σ (A) = 2 1 2 2 3 3 1

Third, from the two–line array σ2(A), i1 i2 ,··· , il σ2(A) = j1 j2 ,··· , jl we construct a pair (S,T) of plane partitions by an insertion and bumping procedure, as follows. The plane partition S will be constructed from i1, i2, ..., il and T from j1, j2, ..., jl. (1) (1) (r) (r) Recursively deﬁne S = i1 and T = j1. Suppose that S and T have been constructed, and these are plane partitions of the same shape, S(r) con- (r) taining the parts i1, i2, ..., ir and T containing j1, j2, ..., jr . (r) We then insert jr+1 into the ﬁrst row of T , immediately to the right of the rightmost entry which is ≥ jr+1. If this space is occupied by some element k, then by entering jr+1 into this space we bump k down to the second row, where it is then treated just as jr+1 was, so that another element may be

12 bumped to the third row, etc. If there is no entry that is ≥ jr+1 then jr+1 is inserted at the beginning of the row and bumps the former ﬁrst element down. (r+1) (r) (r+1) In this way, T is formed from T and jr+1. To construct S just (r) (r+1) insert ir+1 into S so that the resulting array has the same shape as T . ST

3 1

3 2 2 1

3 2 2 2 2 1

3 2 3 2 2 2 1 1

3 2 1 3 3 1 2 1 2 2 1 1 Interchange S and T. Thus the pair of plane partitions which correspond to A is 3 3 1 S = 2 2 1

3 2 1 T = 2 1 1 This completes the third phase of construction. Finally from ordered pair (S,T) of plane partitions, we construct a single plane partition by a method of Frobenius [9], as adapted by Bender and Knuth. From a column of S and a column of T we form a new column, as illustrated below. S • • • 3 • • • 2 • • • 1 T 2 1 0

13 The labels at the right of the array is the first column of S, those at the bottom of the array is one less than the first column of T. By counting the dots in each row, we find the first column of the plane partition, namely

3 3 3

Repeat this with the second column of S and T,

S • • • 3 • • • 2 T 1 0 from which the second column of the plane partition is

3 3

Finally from the third column of S and T, we get S • 1 T 0 and the third column is

Thus the complete partition is

3 3 1 3 3 3

2.3 Basic series and its Combinatorics

Recently in [14], the basic series

∞ n(n+k−1) 2 X q (−q; q )n , (2.2) (q4; q4) n=0 n

14 where k is a positive integer, was interpreted as generating function of two diﬀerent combinatorial objects, viz., an n–color partition function and a weighted lattice path function. The same series has been studied by Agarwal and Rana in [3] using Bender and Knuth matrices. Here in this section, we reproduce the results of [14] and [3]. The following theorem gives the combinatorial interpretation of (2.2) using n–color partition.

Theorem 2.3.1 For a positive integer k, let Ak(ν) denote the number of n–color partitions of ν such that 2.3.1.1 the parts are greater than or equal to k,

2.3.1.2 the parts are of the form (2l − 1)1 or (2l)2, if k is an odd and of the form (2l − 1)2 or (2l)1, if k is an even,

2.3.1.3 if the mi is the smallest or the only part in the partition, then m ≡ i + k − 1(mod 4) and

2.3.1.4 the weighted diﬀerence between any two consecutive parts is non negative and is ≡ 0(mod 4).

Theorem 2.3.2 For k, ν ≥ 1, let Bk(ν) denote the number of BKν–matrices ∆ such that 2.3.2.1 if k is odd (resp. even), then even (resp. odd) columns are zero,

2.3.2.2 all rows after the second row are zero,

th 2.3.2.3 if Ei,j is the (i, j) entry in ∆ such that either it is the only non– zero entry or i + j is minimum, then j ≡ k(mod 4),

2.3.2.4 the order diﬀerence of any two units of ∆ is non negative and is ≡ 0(mod 4),

(k−1) 2.3.2.5 for odd k > 1, the ﬁrst 2 odd columns are zero and

(k−2) 2.3.2.6 for even k > 2, the ﬁrst 2 even columns are zero. Theorem 2.3.3 For all k and ν,

Ak(ν) = Bk(ν)

15 Proof of Theorem 2.3.1 We shall prove that

∞ ∞ n(n+k−1) 2 X X q (−q; q )n A (ν)qν = . (2.3) k (q4; q4) ν=0 n=0 n

Let Ak(m, ν) denote the number of partitions enumerated by Ak(ν) into m parts. We shall prove the identity,

Ak(m, ν) = Ak(m − 1, ν − k − 2m + 2) + Ak(m − 1, ν − k − 4m + 3)

+ Ak(m, ν − 4m). (2.4)

We give the proof of (2.4) for odd k as the proof for even k is similar and hence is omitted. To prove (2.4) for odd k, we split the partitions enumerated by Ak(m, ν) into three classes:

(i) those that have least part equal to k1,

(ii) those that have least part equal to (k + 1)2, and

(iii) those that have least part greater than or equal to (k + 2)1.

We note that in class (iii) the parts are ≥ 51 because if k = 1 then 31 can not be a part in view of the condition (2.3.1.3) of the theorem. We now transform the partitions in class (i) by deleting the least part k1 and then subtracting 2 from all the remaining parts ignoring the subscripts. This produces a partition of ν − k − 2(m − 1) into exactly (m − 1) parts each of which is ≥ k1 (since originally the second smallest part was ≥ (k + 2)1). Obviously, this transformation does not disturb the weighted difference condition (2.3.1.4) between the parts and so the transformed partition is of the type enumerated by Ak(m − 1, ν − k − 2m + 2). Next, we transform the partitions in class (ii) by deleting the least part (k + 1)2 and then subtracting 4 from all the remaining parts. This produces a partition of ν − (k + 1) − 4(m − 1) = ν − k − 4m + 3 into m − 1 parts, each of which is ≥ k1 (since originally the second smallest part was ≥ (k + 4)1). Note that originally (k + 2)1 and (k + 3)2 could not be the second smallest part because of the weighted difference condition (2.3.1.4). Furthermore, since the weighted difference condition between the parts is not disturbed, we see that the transformed partition is of the type enumerated by Ak(m − 1, ν − k − 4m + 3). Finally, we transform the partitions in class (iii) by subtracting 4 from each part ignoring the subscripts. This produces a partition of ν − 4m into m parts, each ≥ k1. Since the weighted difference condition (2.3.1.4) between the parts is again not disturbed, we see that the transformed partition is of

16 the type enumerated by Ak(m, ν − 4m). The above transformations establish a bijection between the partitions enumerated by Ak(m, ν) and those enumerated by Ak(m − 1, ν − k − 2m + 2) + Ak(m − 1, ν − k − 4m + 3) + Ak(m, ν − 4m). This proves the Identity (2.4). For |z| < |q|−1 and |q| < 1, let

∞ ∞ X X m ν fk(z; q) = Ak(m, ν)z q . (2.5) ν=0 m=0

Substituting for Ak(m, ν) from (2.4) in (2.5) and then simplifying, we get the following q–functional equation

k 2 k+1 4 4 fk(z; q) = zq fk(zq ; q) + zq fk(zq ; q) + fk(zq ; q). (2.6)

Since fk(0; q) = 1, we may easily check by coeﬃcient comparison in (2.6) that ∞ n(n+k−1) 2 n X q (−q; q )nz f (z; q) = k (q4; q4) n=0 n Now,

∞ ∞ ∞ X ν X X ν Ak(ν)q = ( Ak(m, ν))q ν=0 ν=0 m=0 = fk(1; q) ∞ n(n+k−1) 2 X q (−q; q )n = (q4; q4) n=0 n This completes the proof of (2.3).

Proof of Theorem 2.3.2 First we deﬁne the mapping denoted by f as

f : Ep,q → (p + q − 1)p, (2.7) and the inverse mapping f −1 is easily seen to be

−1 f : mi → Ei,m−i+1. (2.8)

We shall prove that if ∆ is a matrix enumerated by Bk(ν) then the n–color partition f(∆) is enumerated by Ak(ν), and conversely, if π is an n–color −1 partition enumerated by Ak(ν), then the BKν–matrix f (π) is enumerated by Bk(ν). Let the matrix enumerated by Dk(ν) has the following representation.

∆ = a1,1E1,1 +a1,2E1,2 +···+a2,1E2,1 +a2,2E2,2 +···+a3,1E3,1 +a3,2E3,2 +···

17 Clearly, in view of the condition (2.3.2.4), the entries in ∆ cannot exceed one. Hence, each ai,j = 1 or 0. Let Ep,q,Er,s(p + q ≥ r + s) be two units of ∆ which correspond to two n–color parts mi, nj of f(∆). Then mi = (p+q−1)p and nj = (r + s − 1)r by (2.7). Since (p + q ≥ r + s), therefore m ≥ n and

((mi − nj)) = (p + q − 1) − p − (r + s − 1) − r = q − s − 2r = {{Ep,q − Er,s}} which is non negative and ≡ 0(mod 4). This shows that (2.3.2.4) implies (2.3.1.4). Since f(Ei,j) = (i + j − 1)i = mi (say), by (2.7), so if Ei,j is the only nonzero entry in ∆ or i + j is minimum it means that in f(∆) either mi is the only part or the least part. Thus (2.3.2.3) implies (2.3.1.3). Next, we see that if k is odd, then by (2.3.2.1) even columns in ∆ are zero which means that in Ep,q, q is odd. Further since p ≤ 2 by (2.3.2.2), we conclude that ( q1 if p = 1 f(Ep,q) = (q + 1)2 if p = 2

This shows that in f(∆) the parts are of the form (2l−1)1 or (2l)2. Similarly, we can show that if k is even, then in f(∆) the parts are of the form (2l −1)2 or (2l)1. Thus (2.3.2.1) and (2.3.2.2) imply (2.3.1.2). Finally, when k is odd, say, (2l − 1), the ﬁrst (l − 1) odd columns, that is, 1st, 3rd, ··· , (2l − 3)th are zero by (2.3.2.5) and since E1,2l−3 = (2l − 3)1 and E2,2l−3 = (2l − 2)2, we see that in f(∆) the parts are ≥ k. Thus (2.3.2.5) implies (2.3.1.1) when k is odd. Similarly, we can show that (2.3.2.6) implies (2.3.1.1) when k is even. Thus f(∆) is enumerated by Ak(ν).

To see the reverse implication, let π be an n–color partition of ν enumer- −1 ated by Ak(ν). We shall prove that the BKν–matrix f (π) is enumerated −1 by Bk(ν). Let mi, nj (m ≥ n) be two parts of π such that f (mi) = Ep,q and −1 f (nj) = Er,s. Then Ep,q = Ei,m−i+1 and Er,s = Ej,n−j+1 by (2.8). Since (m ≥ n), we have p + q = m + 1 ≥ n + 1 = r + s, and

{{Ep,q − Er,s}} = {{Ei,m−i+1 − Ej,n−j+1}} = (m − i + 1) − (n − j + 1) − 2j = m − n − i − j

= ((mi − nj))

−1 Thus (2.3.1.4) implies (2.3.2.4) since f (mi) = Ei,m−i+1 = Ei,j (say)(by −1 (2.8)). So if mi is the only part or the least part of π it means that in f (π) either Ei,j is the only non zero entry or i + j is minimum. Thus (2.3.1.3) implies (2.3.2.3). To prove (2.3.2.1), (2.3.2.2), (2.3.2.5) and (2.3.2.6), we ﬁrst consider the case

18 −1 −1 when k is odd. Since f ((2l − 1)1) = E1,2l−1 and f ((2l)2) = E2,2l−1, we see that in f −1(π) even columns are zero and all rows after the second row are zero. This proves (2.3.2.1) and (2.3.2.2). Furthermore, by (2.3.1.1) we −1 −1 see that in f ((2l − 1)1) = E1,2l−1, (2l − 1) ≥ k and in f ((2l)2) = E2,2l−1, (2l) ≥ k, that is (2.3.2.5) and (2.3.2.6) are satisﬁed. Similarly, we can prove the case when k is even. This completes the proof of Theorem 2.3.2. Theorem 2.3.3 leads to a 2–way inﬁnite identity. In [3], for k = 1 and k = 3, the following Rogers–Ramanujan Identities arise as a particular case

∞ ∞ 2 ∞ Y 1 X qn (−q; q2) Y 1 n = ( 4 4 )( n ), (1 − q ) (q ; q )n 1 − q n=1,n≡±1,±2(mod 6) n=0 n=1,n≡±2,±3,6(mod 12) ∞ ∞ 2 ∞ Y 1 X qn +2n(−q; q2) Y 1 n = ( 4 4 )( 4n−2 ). (1 − q ) (q ; q )n 1 − q n=1,n≡±2,3(mod 6) n=0 n=1 The above identities have the following combinatorial interpretations respectively:

Theorem 2.3.4 Let C1(ν) denote the number of partitions of ν into parts ≡ ±2, ±3, 6(mod 12). And let D1(ν) denote the number of partitions into parts ≡ ±1, ±2(mod 6). Then

ν ν X X D1(ν) = A1(k)C1(ν − k) = B1(k)C1(ν − k) k=0 k=0

Theorem 2.3.5 Let C3(ν) denote the number of partitions of ν into parts ≡ 2(mod 4). And let D3(ν) denote the number of partitions of ν into parts ≡ ±2, 3(mod 6). Then

ν ν X X D3(ν) = A3(k)C3(ν − k) = B3(k)C3(ν − k) k=0 k=0 Theorem 2.3.4 and Theorem 2.3.5 lead to a 3–way combinatorial identity

Ak(ν) = Bk(ν) = Ck(ν), k = 1, 3.

2.4 Conclusion

In this chapter, we have explored the bijection between plane partitions and n–color partitions of a number ν. This mapping leads to interesting results as discussed in previous section. The bijection using Bender and Knuth matrices is also used in [18] to prove results on symmetric functions.

19 Chapter 3

On Extensions Of Relations Between Plane Partitions and n–Color Partitions

3.1 Introduction

The purpose of this chapter is to further explore the bijection and ﬁnd some relations between plane partitions and color partitions through Bender and Knuth matrices to establish a simple connection between them. Here, we establish a relation between the number of rows of plane partition and the subscripts of the corresponding n–color partition.

3.2 Main Results

Theorem 3.2.1 Consider an r–rowed plane partition of ν. Then

r = Max[ns] , for s = 1, 2, ··· , t where

m1n1 + m2n2 + m3n3 + ··· + mtnt is an n–color partition of ν.

Proof. Consider an r–rowed plane partition with l columns.

n1,1 n1,2 n1,3 ··· n1,l n2,1 n2,2 n2,3 ··· n2,l ...... nr,1 nr,2 nr,3 ··· nr,l

20 During the ﬁrst phase of conversion, each column of the plane partition is converted to a column of S and T. The entries on the right becomes the column of S and each entry on the bottom is added with one and then becomes the column of T. Converting every column of the plane partition, we get an ordered pair of plane partitions S and T.

S • • • ··· • n1,1 • . . ··· . • . . ··· . . . . ··· . . . . ··· • . • • • T r − 1 Above the Frobenius construction is shown by taking the first column of the plane partition. The first row has n1,1 dots. The first column has r dots, but the first dot is included in the n1,1 dots of the first row. Hence, we are left with r − 1 dots. So, the first entry of first column of T is equal to r. This entry is the maximum entry of T. It can be easily seen that r = Max [entries of T]. S and T are interchanged. Hence r becomes, r = Max [entries of S]. In the second phase of construction, a two–line array is generated from the ordered pair (S,T) such that the entries in the first row are taken from S and the entries in the second row are taken from T. r = Max [Entries of top row of the two–line array].

In the third phase, we obtain Ep,q matrices which give the matrix A. For k > 0, if there are k copies of p in the ﬁrst row and k copies of q in the second row, then ap,q = k.

r = Max [p : ap,q = k > 0].

r = Max [p : Ep,q appears in the representation of matrix A].

The row number p in Ep,q is nothing but the subscript ns if msns is the corresponding part of Ep,q matrix in the n–colour partition of ν. Hence,

r = Max [ns] where 1 ≤ s ≤ t.

21 Theorem 3.2.2 If mi is the only part in an n–color partition of ν, then the correponding plane partition has i rows, 1 column, m − i + 1 occurs as an entry in the ﬁrst row and the rest i − 1 entries are all 1’s.

Proof. Since mi is the only part, there is only one Ep,q matrix which is Ei,m−i+1. As ai,m−i+1 = 1 > 0 , there is one copy of i in the ﬁrst row and one copy of m − i + 1 in the second row of the two–line array. The two–line array is i m − i + 1 The ordered pair of plane partitions (S,T) after interchanging becomes

ST m − i + 1 i

By the method of Frobenius, taking entries of S on the right and one less than entries of T on the bottom, we get

S • • • ··· • m − i + 1 • • . . • T i − 1 The first entry of S is m − i + 1, hence adding m − i + 1 dots in the first row. The first entry of T is i − 1, so creating i − 1 rows with one dot in each row. The corresponding plane partition can be obtained by counting the number of dots in each row. Hence, we obtain

m − i + 1 1 1 . . 1

It has i rows, 1 column, m − i + 1 occurs as an entry in the ﬁrst row and the rest i − 1 entries are all 1’s.

22 Remark 3.2.1 If we have t parts in an n–color partition of ν,

m1n1 + m2n2 + m3n3 + ··· + mtnt where ms ≥ ms+1 , ns ≥ ns+1 , ms −ns +1 ≥ ms+1 −ns+1 +1 ∀ 1 ≤ s ≤ t−1 then converting each part using Theorem 3.2.2 to a column of plane partition we get the corresponding plane partition.

m1 − n1 + 1 m2 − n2 + 1 . . . mt − nt + 1 1 1 ... 1 .... 1 .. 1 1 1 1 1

Since n1 is the largest subscript, there are n1 rows in the plane partition. Each part of the n–color partition corresponds to one column in this case, hence, there are t columns. The ﬁrst column has n1 − 1 entries equal to one, second column has n2 − 1 entries equal to one, and so on.

Corollary 3.2.3 If

m11 + m21 + m31 + ··· + mt1 where ms ≥ ms+1 ∀ s ≥ 1 is an n–color partition of ν, then the corresponding plane partition is m1 m2 m3 ··· mt in 1 row and t columns.

Proof. Since ns = 1 ∀ s ≥ 1, ns ≥ ns+1 is satisﬁed. Also ms ≥ ms+1 ∀s. Since ms − 1 + 1 = ms, ms − ns + 1 ≥ ms+1 − ns+1 + 1 is also satisﬁed. Hence using Theorem 3.2.2 and Remark 3.2.1, the corresponding plane partition is m1 m2 m3 ··· mt

Corollary 3.2.4 If

m1m1 + m2m2 + m3m3 + ··· + mtmt with ms ≥ ms+1 is an n–color partition of ν, then the corresponding plane partition has m1 rows, t columns with all entries equal to one.

23 Proof. Since ms ≥ ms+1 ∀ s ≥ 1, all the three conditions stated in Re- mark 3.2.1 are satisiﬁed. Hence, by Theorem 3.2.2 and Remark 3.2.1, the corresponding plane partition is 1 1 ... 1 1 1 ... 1 .... 1 ... 1 1 1 1

Since m1 is the largest subscript of all the parts of the n–color partition, the number of rows in the plane partition is m1. As ms − ns + 1 = 1 ∀s = 1, 2, ··· , t, all the entries of the ﬁrst row are equivalent to one. The ﬁrst column has m1 number of 1’s, the second column th has m2 number of 1’s, ··· , the t column has mt number of 1’s.

Theorem 3.2.5 If m1 + m2 + m3 + ··· + mt−1 + mt with t ≤ m is an n– color partition of ν, then the corresponding plane partition has t rows and one column, with each entry equal to m.

Proof. Let m1 + m2 + m3 + ··· + mt−1 + mt be an n–color partition of ν, then the matrix A can be written as sum of Ep,q matrices as

A = E1,m + E2,m−1 + E3,m−2 + ··· + Et−1,m−t+2 + Et,m−t+1 The two–rowed array becomes 1 2 3 ··· t − 1 t σ (A) = 1 m m − 1 m − 2 ··· m − t + 2 m − t + 1 The entries of the top row should be in non-increasing order. t t − 1 ··· 3 2 1 σ (A) = 2 m − t + 1 m − t + 2 ··· m − 2 m − 1 m The ordered pair (S,T) of plane partitions becomes TS

t m t − 1 m − 1 . . . . 3 m − t + 3 2 m − t + 2 1 m − t + 1 By Frobenius construction,

24 S • • • ··· • m • • • ··· • m − 1 • • • ··· • m − 2 ...... ··· . . • • • ··· • m − t + 1 T t − 1 t − 2 t − 3 ··· 0 Each row has m dots. The plane partition is

m m m . . m having t rows and a single column.

3.3 Applications of Plane partitions

3.3.1 Diamond Partitions Diamond Partitions were introduced by G.E. Andrews and P. Paule in [7] as new variations of plane partitions in 2001. The culmination of their study leads to an inﬁnite family of modular forms. These, in turn, lead to interesting arithmetic theorems and conjectures for the related partition functions. k–Elongated Partition Diamond of length n

There are (2k + 1)n + 1 parts where n is the length and k is the elongation in diamond.

25 Its generating function is

Qn−1 (2k+1)j+2 (2k+1)j+4 (2k+1)j+2k j=0 (1 + q )(1 + q )...... (1 + q ) hn,k = Q(2k+1)n+1 j j=1 (1 − q ) A diamond partition can be obtained from a plane partition only when the plane partition is of the following forms:

1. 2 × 2 A 2 × 2 plane partition a11 a12 a21 a22 can be written as a 1–elongated diamond partition of length one.

2. m × 2 A m × 2 plane partition a11 a12 a21 a22 a31 a32 . . . . am1 am2 can be written as a m − 1 elongated diamond partition

26 if the plane partition satisﬁes some additional conditions:

a12 ≥ a31

a22 ≥ a41

a32 ≥ a51 . . . .

a(m−2)2 ≥ am1

3. 2 × n A 2 × n plane partition

a11 a12 a13 ··· a1n a21 a22 a23 ··· a2n can be written as a n − 1 elongated diamond partition

if the plane partition satisiﬁes some additional conditions:

a21 ≥ a13

a22 ≥ a14

a23 ≥ a15 . . . .

a2(n−2) ≥ a1n

3.3.2 Solid Partitions In mathematics, solid partitions are natural generalizations of partitions and plane partitions deﬁned by Percy Alexander MacMahon in [17]. A solid partition of n is a three–dimensional array, ni,j,k, of non–negative integers (the indices i, j, k ≥ 1) such that X ni,j,k = n i,j,k

27 and

ni,j,k ≥ n(i+1),j,k , ni,j,k ≥ ni,(j+1),k , ni,j,k ≥ ni,j,(k+1) , ∀i, j, k.

As the deﬁnition of solid partitions involves three–dimensional arrays of numbers, they are also called three–dimensional partitions in notation where plane partitions are two–dimensional partitions and partitions are one–dimensional partitions. There is a one to one correspondence between plane partitions and solid partitions in which the part magnitude is limited by unity. If we take a plane partition in the xy–plane, we can obtain the corresponding solid partition by replacing each part by a pile of nodes in the direction of z–axis. The plane partition arises by projection of the solid partition upon one of the coordinate planes.

28 Bibliography

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